A187204 -id:A187204 - cp's OEIS frontend

A187202 The bottom entry in the difference table of the divisors of n.

1, 1, 2, 1, 4, 2, 6, 1, 4, 0, 10, 1, 12, -2, 8, 1, 16, 12, 18, -11, 8, -6, 22, -12, 16, -8, 8, -3, 28, 50, 30, 1, 8, -12, 28, -11, 36, -14, 8, -66, 40, 104, 42, 13, 24, -18, 46, -103, 36, -16, 8, 21, 52, 88, 36, 48, 8, -24, 58, -667, 60, -26, -8, 1, 40, 72

Offset: 1

Omar E. Pol, Aug 01 2011

Note that if n is prime then a(n) = n - 1.
Note that if n is a power of 2 then a(n) = 1.
a(A193671(n)) > 0; a(A187204(n)) = 0; a(A193672(n)) < 0. [Reinhard Zumkeller, Aug 02 2011]
First differs from A187203 at a(14). - Omar E. Pol, May 14 2016
From David A. Corneth, May 20 2016: (Start)
The bottom of the difference table of the divisors of n can be expressed in terms of the divisors of n and use of Pascal's triangle. Suppose a, b, c, d and e are the divisors of n. Then the difference table is as follows (rotated for ease of reading):
a
. . b-a
b . . . . c-2b+a
. . c-b . . . . . d-3c+3b-a
c . . . . d-2c+b . . . . . . e-4d+6c-4b+a
. . d-c . . . . . e-3d+3c-b
d . . . . e-2d+c
. . e-d
e
From here we can see Pascal's triangle occurring. Induction can be used to show that it's the case in general.
(End)

			a(18) = 12 because the divisors of 18 are 1, 2, 3, 6, 9, 18, and the difference triangle of the divisors is:
1 . 2 . 3 . 6 . 9 . 18
. 1 . 1 . 3 . 3 . 9
. . 0 . 2 . 0 . 6
. . . 2 .-2 . 6
. . . .-4 . 8
. . . . . 12
with bottom entry a(18) = 12.
Note that A187203(18) = 4.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000005, A007318, A027750, A187203, A273102.

a187202 = head . head . dropWhile ((> 1) . length) . iterate diff . divs
   where divs n = filter ((== 0) . mod n) [1..n]
         diff xs = zipWith (-) (tail xs) xs
-- Reinhard Zumkeller, Aug 02 2011

f:=proc(n) local k,d,lis; lis:=divisors(n); d:=nops(lis);
add( (-1)^k*binomial(d-1,k)*lis[d-k], k=0..d-1); end;
[seq(f(n),n=1..100)]; # N. J. A. Sloane, May 01 2016

Table[d = Divisors[n]; Differences[d, Length[d] - 1][[1]], {n, 100}] (* T. D. Noe, Aug 01 2011 *)

A187202(n)={ for(i=2,#n=divisors(n), n=vecextract(n,"^1")-vecextract(n,"^-1")); n[1]}  \\ M. F. Hasler, Aug 01 2011

a(n) = Sum_{k=0..d-1} (-1)^k*binomial(d-1,k)*D[d-k], where D is a sorted list of the d = A000005(n) divisors of n. - N. J. A. Sloane, May 01 2016

a(2^k) = 1.

Edited by N. J. A. Sloane, May 01 2016

A273102 Difference table of the divisors of the positive integers.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 2, 1, 2, 4, 1, 2, 1, 1, 5, 4, 1, 2, 3, 6, 1, 1, 3, 0, 2, 2, 1, 7, 6, 1, 2, 4, 8, 1, 2, 4, 1, 2, 1, 1, 3, 9, 2, 6, 4, 1, 2, 5, 10, 1, 3, 5, 2, 2, 0, 1, 11, 10, 1, 2, 3, 4, 6, 12, 1, 1, 1, 2, 6, 0, 0, 1, 4, 0, 1, 3, 1, 2, 1, 1, 13, 12, 1, 2, 7, 14, 1, 5, 7, 4, 2, -2, 1, 3, 5, 15, 2, 2, 10, 0, 8, 8

Offset: 1

Omar E. Pol, May 15 2016

This is an irregular tetrahedron T(n,j,k) read by rows in which the slice n lists the elements of the rows of the difference triangle of the divisors of n (including the divisors of n).
The first row of the slice n is also the n-th row of the triangle A027750.
The bottom entry of the slice n is A187202(n).
The sum of the elements of the slice n is A273103(n).
For another version see A273104, from which differs at a(92).
From David A. Corneth, May 20 2016: (Start)
Each element of the difference table of the divisors of n can be expressed in terms of the divisors of n and use of Pascal's triangle. Suppose a, b, c, d and e are the divisors of n. Then the difference table is as follows (rotated for ease of reading):
a
. . b-a
b . . . . c-2b+a
. . c-b . . . . . d-3c+3b-a
c . . . . d-2c+b . . . . . . e-4d+6c-4b+a
. . d-c . . . . . e-3d+3c-b
d . . . . e-2d+c
. . e-d
e
From here we can see Pascal's triangle occurring. Induction can be used to show that it's the case in general.
(End)

			For n = 18 the divisors of 18 are 1, 2, 3, 6, 9, 18, so the difference triangle of the divisors of 18 is
  1 . 2 . 3 . 6 . 9 . 18
    1 . 1 . 3 . 3 . 9
      0 . 2 . 0 . 6
        2 .-2 . 6
         -4 . 8
           12
and the 18th slice is
  1, 2, 3, 6, 9, 18;
  1, 1, 3, 3, 9;
  0, 2, 0, 6;
  2,-2, 6;
  -4, 8;
  12;
The tetrahedron begins:
  1;
  1, 2;
  1;
  1, 3;
  2;
  1, 2, 4;
  1, 2;
  1;
  ...
This is also an irregular triangle T(n,r) read by rows in which row n lists the difference triangle of the divisors of n flattened. Row lengths are the terms of A184389. Row sums give A273103.
Triangle begins:
  1;
  1, 2, 1;
  1, 3, 2;
  1, 2, 4, 1, 2, 1;
  ...

Cf. A007182, A027750, A184389, A187202, A187204, A273103, A273104, A273109.

Table[Drop[FixedPointList[Differences, Divisors@ n], -2], {n, 15}] // Flatten (* Michael De Vlieger, May 16 2016 *)

def A273102_DTD(n): # DTD = Difference Table of Divisors
    D = divisors(n)
    T = matrix(ZZ, len(D))
    for (m, d) in enumerate(D):
        T[0, m] = d
        for k in range(m-1, -1, -1) :
            T[m-k, k] = T[m-k-1, k+1] - T[m-k-1, k]
    return [T.row(k)[:len(D)-k] for k in range(len(D))]
# Keeps the rows of the DTD, for instance
# A273102_DTD(18)[1] = 1,1,3,3,9 (see the example above).
for n in range(1,19): print(A273102_DTD(n)) # Peter Luschny, May 18 2016

A187205 Numbers such that the last of the absolute differences of divisors is 0.

Original entry on oeis.org

10, 40, 50, 56, 104, 130, 136, 160, 170, 171, 224, 230, 232, 250, 290, 310, 312, 370, 392, 410, 430, 459, 470, 520, 530, 560, 590, 610, 624, 640, 648, 670, 710, 730, 790, 830, 890, 896, 970, 1000, 1010, 1030, 1070, 1088, 1090, 1130, 1160, 1216, 1218, 1221

Offset: 1

Omar E. Pol, Aug 01 2011

nonn

Numbers n such that A187203(n) = 0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A027750, A187202, A187203, A187204.

import Data.List (elemIndices)
a187205 n = a187205_list !! (n-1)
a187205_list = map (+ 1) $ elemIndices 0 $ map a187203 [1..]

A187208 Numbers such that the last of the absolute differences of divisors is 1.

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 20, 32, 36, 48, 64, 80, 108, 112, 128, 156, 192, 204, 220, 252, 256, 260, 272, 304, 320, 324, 368, 396, 448, 476, 484, 512, 544, 608, 656, 660, 688, 768, 972, 1008, 1024, 1044, 1120, 1184, 1248, 1280, 1300, 1332, 1476, 1764, 1792, 1908

Offset: 1

Omar E. Pol, Aug 01 2011

nonn

Numbers n such that A187203(n) = 1.

A000079 is a subsequence (powers of 2). [Reinhard Zumkeller, Aug 02 2011]

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A027750, A187202, A187203, A187204, A187205.

import Data.List (elemIndices)
a187208 n = a187208_list !! (n-1)
a187208_list = map (+ 1) $ elemIndices 1 $ map a187203 [1..]
-- Reinhard Zumkeller, Aug 02 2011

lad1Q[n_]:=Nest[Abs[Differences[#]]&,Divisors[n],DivisorSigma[0,n]-1]=={1}; Select[Range[2000],lad1Q] (* Harvey P. Dale, Nov 07 2022 *)

A193672 Numbers such that the last of the differences of divisors is < 0.

Original entry on oeis.org

14, 20, 22, 24, 26, 28, 34, 36, 38, 40, 46, 48, 50, 58, 60, 62, 63, 70, 74, 80, 82, 84, 86, 94, 96, 98, 99, 100, 105, 106, 117, 118, 120, 122, 134, 136, 138, 140, 142, 146, 152, 153, 154, 158, 160, 166, 170, 174, 178, 180, 182, 184, 186, 189, 190, 192, 194

Offset: 1

Reinhard Zumkeller, Aug 02 2011

nonn

A187202(a(n)) < 0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A187204, A193672.

import Data.List (findIndices)
a193672 n = a193672_list !! (n-1)
a193672_list = map (+ 1) $ findIndices (< 0) $ map a187202 [1..]

Select[Range[200],Differences[Divisors[#],DivisorSigma[0,#]-1][[1]]<0&] (* Harvey P. Dale, Feb 14 2025 *)

A273261 Irregular triangle read by rows: T(n,k) = sum of the elements of the k-th row of the difference table of the divisors of n.

Original entry on oeis.org

1, 3, 1, 4, 2, 7, 3, 1, 6, 4, 12, 5, 2, 2, 8, 6, 15, 7, 3, 1, 13, 8, 4, 18, 9, 4, 0, 12, 10, 28, 11, 5, 4, 3, 1, 14, 12, 24, 13, 6, -2, 24, 14, 8, 8, 31, 15, 7, 3, 1, 18, 16, 39, 17, 8, 6, 4, 12, 20, 18, 42, 19, 9, 4, 3, -11, 32, 20, 12, 8, 36, 21, 10, -6, 24, 22, 60, 23, 11, 8, 6, 3, 4, -12, 31, 24, 16, 42, 25, 12, -8

Offset: 1

Omar E. Pol, May 20 2016

Row 2^k gives the first k+1 positive terms of A000225 in decreasing order, k >= 0.
If n is prime then row n contains only two terms: n+1 and n-1.
First differs from A274531 at a(41).
For n = p^k, T(n, 1) = n - 1, T(n, n) = (p - 1)^k. a(A006218(n - 1) + 1) = T(n, 0), a(A006218(n)) = T(n, t-1) where t is the number of divisors of n. - David A. Corneth, Jun 18 2016
Let D_n(m, c) be the k-th element in row m. The divisors of n are in row m = 0. Let t be the number of divisors of n. Then T(n, k) = D_n(k - 1, t-1) - D_n(k - 1, 0). - David A. Corneth, Jun 25 2016
For n in A187204, the last term of the n-th row is 0. - Michel Marcus, Apr 02 2017

			Triangle begins:
1;
3, 1;
4, 2;
7, 3, 1;
6, 4;
12, 5, 2, 2;
8, 6;
15, 7, 3, 1;
13, 8, 4;
18, 9, 4, 0;
12, 10;
28, 11, 5, 4, 3, 1;
14, 12;
24, 13, 6, -2;
24, 14, 8, 8;
31, 15, 7, 3, 1;
18, 16;
39, 17, 8, 6, 4, 12;
20, 18;
42, 19, 9, 4, 3, -11;
32, 20, 12, 8;
36, 21, 10, -6;
24, 22;
60, 23, 11, 8, 6, 3, 4, -12;
31, 24, 16;
42, 25, 12, -8;
...
For n = 14 the divisors of 14 are 1, 2, 7, 14, and the difference triangle of the divisors is
1, 2, 7, 14;
1, 5, 7;
4, 2;
-2;
The row sums give [24, 13, 6, -2] which is also the 14th row of the irregular triangle.
In the first row, the last element is 14, the first is 1. So the sum of the second row is 14 - 1 is 13. Similarly, the sum of the third row is 7 - 1 = 6, and of the last row, 2 - 4 = -2. - _David A. Corneth_, Jun 25 2016

Row lengths give A000005. Column 1 is A000203.
Right border gives A187202. Row sums give A273103.
Cf. A000225, A006218, A187204, A273102, A273262, A273263, A274531.

Map[Total, Table[NestWhileList[Differences, Divisors@ n, Length@ # > 1 &], {n, 26}], {2}] // Flatten (* Michael De Vlieger, Jun 26 2016 *)

row(n) = {my(d = divisors(n));my(nd = #d); my(m = matrix(#d, #d)); for (j=1, nd, m[1,j] = d[j];); for (i=2, nd, for (j=1, nd - i +1, m[i,j] = m[i-1,j+1] - m[i-1,j];);); vector(nd, i, sum(j=1, nd, m[i, j]));}
tabf(nn) = for (n=1, nn, print(row(n)););
lista(nn) = for (n=1, nn, v = row(n); for (j=1, #v, print1(v[j], ", "));); \\ Michel Marcus, Jun 25 2016

A193671 Numbers such that the last of the differences of divisors is > 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 21, 23, 25, 27, 29, 30, 31, 32, 33, 35, 37, 39, 41, 42, 43, 44, 45, 47, 49, 51, 52, 53, 54, 55, 56, 57, 59, 61, 64, 65, 66, 67, 68, 69, 71, 72, 73, 75, 76, 77, 78, 79, 81, 83, 85, 87, 88, 89, 90, 91

Offset: 1

Reinhard Zumkeller, Aug 02 2011

nonn

A187202(a(n)) > 0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A187204, A193672.

import Data.List (findIndices)
a193671 n = a193671_list !! (n-1)
a193671_list = map (+ 1) $ findIndices (> 0) $ map a187202 [1..]

A272374 Numbers n such that A187202(n) is <= 0.

Original entry on oeis.org

10, 14, 20, 22, 24, 26, 28, 34, 36, 38, 40, 46, 48, 50, 58, 60, 62, 63, 70, 74, 80, 82, 84, 86, 94, 96, 98, 99, 100, 105, 106, 117, 118, 120, 122, 134, 136, 138, 140, 142, 146, 152, 153, 154, 158, 160, 166, 170, 171, 174, 178, 180, 182, 184, 186, 189, 190, 192, 194, 196, 198, 200, 202, 206, 208, 214

Offset: 1

Robert G. Wilson v, Apr 28 2016

nonn

Odd terms: 63, 99, 105, 117, 153, 171, 189, ..., .

Indices n where A187202(n) =0 are 10, 171, 1947, 2619, 265105, ...- R. J. Mathar, May 06 2016

R. J. Mathar, Table of n, a(n) for n = 1..5000

Cf. A187202, A187204, A272375.

f[n_] := Block[{d = Divisors@ n}, Differences[d, Length@ d - 1][[1]]]; Select[ Range@ 215, f@# < 1&]

Edited by N. J. A. Sloane, May 01 2016

cp's OEIS Frontend

A187202 The bottom entry in the difference table of the divisors of n.

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A273102 Difference table of the divisors of the positive integers.

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A187205 Numbers such that the last of the absolute differences of divisors is 0.

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A187208 Numbers such that the last of the absolute differences of divisors is 1.

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A193672 Numbers such that the last of the differences of divisors is < 0.

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A273261 Irregular triangle read by rows: T(n,k) = sum of the elements of the k-th row of the difference table of the divisors of n.

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A193671 Numbers such that the last of the differences of divisors is > 0.

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A272374 Numbers n such that A187202(n) is <= 0.

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